~RD-Ai48 478 COHERENT STRUCTURE MODELING OF VISCOUS SUBLAYER i/I'A ITIURBULENCE FOR INCOMPRESS. -(U) STANFORD UNIV CA DEPT OF

AERONAUTICS AND ASTRONAUTICS D K OTA ET AL- FEB 84

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D!psu tmmmt 4of AERONAUTICS anud ASTRONAUTICS-STANMFORDI UNIVIERSITVST NO DU.VRU Report AA CFD 84-1

SECOND ANNUAL SCIENTIFIC REPORT

January 1, 1983 to December 31, 1983

Air Force Office of Scientific Research Contract 81-NA-256

COIIRFN'I STRUCTURE MODELING OF VISCOUS SUBLAYER TURBULENCE

FOR INCOMPRESSIBLE FLOW WITH HEAT TRANSFER

9AND FOR COMPRESSIBLE FLOW

BY

DALE K. OTA AND DEAN R. CHAPMAN

Submitted to the

Directorate of Aerospace Sciences

Air Force Office of Scienitific Research

Boiling AFB "

Washington D.C. 20332

C-:L. by the -,

Dep.trtrnent of Aeronautics and AstronauticsLad

Stanford UniversityLL.

CZ Stanford, CA 94305-2186

February 1984 4

R 4 "" "

k-7~

- Report AA CFD 84-1

SECOND ANNUAL SCIENTIFIC REPORT

January 1, 1983 to December 31, 1983

Air Force Office of Scientific Research Contract 81-NA-256

COHERENT STRUCTURE MODELING OF VISCOUS SUBLAYER TURBULENCE

FOR INCOMPRESSIBLE FLOW WITH HEAT TRANSFER

AND FOR COMPRESSIBLE FLOW

BY

DALE K. OTA AND DEAN R. CHAPMAN**a W

Submitted to the

Directorate of Aerospace Sciences

Air Force Office of Scientific Research

Boiling AFB

Washington D.C. 20332

by the

Department of Aeronautics and Astronautics

Stanford University

Stanford, CA 94305-2186

February 1984

* . . . . .. * *.... ..a -, -' .... *-,.T *.T r .* ... .

IJ',T F T

P P'-

SFCURITY CLASSIFICATION OF THIS PAGE (T.n Data Entered)

READ INSTRUCTIONSREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORMI. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBERAF -"i 34-0248 74. TITLE (ad Subtitle) S. TYPE OF REPORT & PERIOD COVERED

second Annual Scientific Rpt.Coherent Sturcture Modeling,,of Viscous Sublayer Ja.1193tDe.1,98Turbulence for Incompressible Flow with Heat Jan. 1, 1983 to Dec. 31, 1983

Transfer and for Compressible Flow 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(s) I. CONTRACT OR GRANT NUMBER(s)

Dale K. Ota and Dean R. Chapman AFOSR 82-0083

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT. TASKAREA 6 WORK UNIT NUMBERSDepartment of Aeronautics and Astronautics ARE OR U M(ii1o. r~Stanford University AO IAZ.Stanford, CA 94305 -2It

I I. CONTROLLING OFFICE NAME AND ADDRESS ,2. REPORT DATEDirectorate of Aerospace Sciences L J February 1984Air Force Office of Scientific Research 13. NUMBER OF PAGES

Bolling AFB, Washington, D.C. 20332 _

14. MONITORING AGENCY NAME & ADDRESS(f different from Controlling Office) IS. SECURITY CLASS. (of this report)

ISa, DECL ASSI FI C ATION/DOWN GRADIN GSCHEDULE

1. DISTRIBUTION STATEMENT (of this Report)

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, It different from Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue ot reverse side it necessary and identify by block numbcr)

Fluid Mechanics, heat transfer, turbulent Prandtl number, viscous sublayer,Navier-Stokes, Computational Model

20. ABSTEIACT (Continue on reverse side If necessary end Identify b block n,,mber)

-'" The general objective of the present research is to develop a"""" Navier-Stokes computational model of the time-dependent dynamics and

" ]heat transfer in a compressible viscous sublayer. The main objectiveis to compute the variation of turbulent Prandtl number across thesublayer. Experiments have been unable to define this variation, andexisting theories differ greatly.-''-- " (continued on reverse)

DD I JAN 73 1473SECURITY CLASSIFICATION OF- THIS PAGE -l e-n Deta Potrel' )

. ' "9 . , " ' , , . - . v - . . ' - . " , ., , ' . .

"~~hFIROSECURITY CLASSIFICATION OF THIS PAGE(When Data Enterd)

(20. continued)

--- A computational code has been developed using preliminary,relatively simple temperature and velocity boundary conditions atthe outer edge of the sublayer. Computations have been made formolecular Prandtl numbers from 0.7 to 6 with zero pressure gradient,and for adverse, zero, and favorable pressure gradient, with aPrandtl number of 0.7. These preliminary results show a strongeffect of molecular Prandtl number on turbulent Prandtl numbernear the wall; but only a relatively small effect of pressuregradient throughout the sublayer. Future computations will bemade with more refined boundary conditions.

~Accec,

.ST

I -..• ,.

-S'SEUIY LSIFCTO O HS A,.enDt Ettd

I .I .* -

INTRODUCTION AND OBJECTIVES

Practical applications of computational fluid dynamics within the foreseeable future

must necessarily utilize some form of turbulence modeling. Computations based on the

." Reynolds-averaged equations of motion require all turbulence transport of momentum and

energy to be modeled; while large-eddy simulations require the subgrid-scale turbulence

to be modeled. In both cases, the uncertainties in modeling turbulence within the viscous

sublayer constitute a major weak link in the overall numerical computation.

During the past two decades a great deal of new experimental information has been

" " assembled on the physics of organized eddy structures in turbulent flow, especially within

the viscous sublayer (see, for example, the recent review of Cantwell, 1981). Yet it has not

been possible thus far to incorporate this body of physical information within the frame-

work of Reynolds-averaged turbulence modeling. The reason is fundamental, reflecting a

well-known limitation in the Reynolds-average approach which begins by time averaging

the dynamic equations of motion. In this initial mathematical step important physical

aspects of organized eddy structures, such as phase relationships and coherent structure-%.

"" '"dynamics, are obliterated irreversibly. Consequently, some totally different approach is

required if the observed physics of coherent eddy motions are to be incorporated within

the framework of a turbulence model.

Quite recently a new approach has been developed for modeling viscous sublayer tur-

bulence in incompressible flow without heat transfer (Chapman and Kuhn (1981). Their

":-' method models directly the essential organized eddy structures observed in experiments.

"0' The principal steps in this "coherent-structure modelling" are: first, to model velocity

boundary conditions at the outer edge of the viscous sublayer, then to compute time-

dependent dynamics, and finally to time average computed results. Thus, time averaging

o 'is the last operation performed on computed dynamics, rather than the first operation

performed on the dynamic equations. This initial effort, although far from fully devel-

oped, already has been surprisingly successful in modeling some important characteristics

' . o,' ' " ° e ,- '° ° ° " ° . . . . , % ' ' o% % e % , ". ". " • . - - - - ",°- - . % , , . - •2

of viscous sublayer turbulence.

The over all objective of the present research is to develop, using the Navier-Stokes

equations, a computational model of viscous sublayer turbulence applicable to flow with

heat transfer and to compressible flow. Specific tasks within this objective are to utilize

the model to compute the distribution of turbulent Prandtl number within the viscous

sublayer (a) for fluids of various molecular Prandtl number, and (b) for flows with adverse,

zero, and favorable streamwise pressure gradients.

Navier-Stokes computations of the turbulent Prandtl number Pr are significant be-

cause experimental techniques have been unable to provide reliable measurements within

the viscous sublayer. This parameter, of course, affects turbulent heat transfer. Since by

definition,

Pri-

an experimental determination would require accurate measurements near a wall of the

Reynolds stress V, the correlation Ov between temperature 0 and normal velocity, the

mean velocity gradient au/Ty, and the mean temperature gradient a/Oay. To date, it has

not been possible to make reliable measurements of these four quantities in the viscous

sublayer. The band of uncertainty from one set of data for air (Simpson et al. (1970)),

is illustrated in fig. 1. Other sets of experimental data, e.g. Fulachier (1972), indicate

that the overall uncertainty band is even broader than this one set of measurements would

indicate. Experiments are unable to determine whether the values of Pr near a wall are

large, small, or intermediate. Experiments also have not been able to define how Pr varies

with molecular Pr or with pressure gradient within the viscous sublayer. We believe that

Navier-Stokes computations can shed much light on these uncertainties.

. ..

; c.'

MOTIVATIONS FOR COHERENT STRUCTURE MODELING OF

VISCOUS SUBLAYER TURBULENCE

Coherent-structure modeling departs markedly from Reynolds-averaging modeling,

and thereby offers the potential of some entirely new advances in turbulence computation.

Both the motivations and the payoff for this type of research are quite different from those

for more conventional turbulence modeling. In addition to the specific motivations outlined

above for the objectives of the present research, there are other important motivations that

will affect different areas of turbulence computation. It is appropriate, therefore, to outline

some of these.

Three other motivations for developing a realistic coherent-structure model of viscous

sublayer turbulence are:

'1 To provide a basis for strengthening a major weak link in present Reynolds-average

closure schemes. Conventional turbulence closure methods (e.g. k-C, other 2-equation

methods, and Reynolds-Stress transport methods) all employ differential transport

equations that are "modeled" forms of exact transport equations. The exact equation

for the transport of dissipation, for example, contains very complex turbulence terms

involving pressure-velocity correlations, triple correlations of velocity gradients, cor-

prelations of second derivatives of velocity, and correlations between pressure gradient

and velocity gradient. Such correlations are not measurable with present experimental

technology. Because modeled equations for free turbulence yield demonstrably incor-

rect results near a wall, various ad hoc functions (up to 5 in number) are added in an

effort to mend this shortcoming. Without any guide or test basis from experiment, the

inevitable consequence is that different models with different ad hoc functions yield

different results for technically important quantities such as skin friction in flows with

pressure gradient (Patel et. al 1981). If, however, a realistic Navier-Stokes computa-

tional model were developed for the time dependent dynamics of viscous sublayer flow,

then all the terms in the exact transport equations could be computed. When compared

.1

- . . . . . ....

with corresponding modeled terms, a rational formulation would be obtained of the wall

4"" damping functions for integrating turbulence models across the viscous sublayer. This

. "would strengthen a weak link in conventional turbulence models. Reynolds-average

modeling, of course, will be the mainstay of practical turbulent flow computation for

" "years to come.

2 To provide a simple test flow against which sub grid scale models of turbulence in large

eddy simulations can be tested. The simple flow of homogeneous shear, for example,

computed in detail from the time-dependent Navier-Stokes equations (Rogallo 1977)

has been used effectively as a test flow for assessing subgrid scale turbulence models

for application in the outer region of a turbulent boundary layer or in free shear layers

(e.g., Clark et. al. 1979, McMillan and Ferziger 1979, Shirani 1981). A realistic

coherent-structure model of viscous sublayer turbulence, likewise computed from the

time-dependent Navier-Stokes equations, could similarly be used to test subgrid-scale

models for large eddy simulations of turbulent flow in the region adjacent to a wall.

3 - To provide a guide for modeling the lower boundary conditions for the outer turbulent

region in large eddy simulations of boundary layer flow. If a realistic boundary con-

dition of this type could be developed, the entire viscous sublayer could be modeled

-.. .-. rather than directly computed. The required computer power for large eddy simula-

tions at high Reynolds numbers would thereby be reduced by a very large factor (about

103; Chapman, 1980).

In summary of the motivations for developing coherent-structure turbulence models,

it is clear that the potential payoffs are significant and of much broader scope than the

specific objectives of the present research.

.". .°.•

. . . . . .. . . . . . . . . . . . . . . . . . .

,% %*

FORMULATION OF COMPUTATIONAL MODEL

DIFFERENTIAL EQUATIONS

The numerical scheme selected to solve the Navier-Stokes equations is an adaptation

of the Pulliam and Steger (1980) code for compressible flow. Velocities in the physical z,

% y domain are u, i, I. In the computational , q7, , domain the dependent variables

are p, pu, pr, pi, and p [ei + (U2 + V2 + W2) /2], where ei is the internal energy per unit

mass. Pulliam and Steger use 2 as the coordinate normal to the surface, and y as the

spanwise coordinate. In order to minimize code changes, we use this coordinate system in

the numerical computations and then, at the end, change 2 to y, W to v, y to z, and U to w

in order to present results in the more familiar boundary layer coordinate designations. We

employ simple cartesian coordinates in the streamwise z- and spanwise z = Y-directions,

and a stretched mesh in the normal y = direction in order to concentrate points both near

the wall and the outer edge. The metrics for the transformation are i 1, = Ay+,

I = 1/Ak+ where Ay+ and A2+ are the mesh spacings in wall units in the spanwise and

normal directions respectively.

The same basic physical approximation is made as in Chapman and Kuhn (1981),

namely, that coherent sublayer eddy structures are highly elongate streamwise. In our

case we assume-in accordance with experimental observations (e.g. Iritani et. al (1981))

- that the temperature eddies are also highly elongate streamwise. In the notation of

Pulliam and Steger, the corresponding divergence form of the Navier-Stokes equations is

.o.: 0O - aFe -F) t- G,, - G) + Fp, 14,

where the vectors F, F., G, GQ and are identical to the conventional ones used by Pulliam

and Steger for the case of orthogonal coordinates corresponding to the metrics of equation

°%' .%

..................... '''q.r -,-2.,R.I:l.A.~'*

(1). However, the vector

0

ON

-j-"[U8EC-+ i2(PW)]

differs from theirs due to our assumption of highly elongate eddies in the streamwise

. direction. Thus, streamwise derivatives other than the mean streamwise pressure gradient

dp./dx are neglected relative to derivatives in the spanwise and normal directions.

This basic approximation, which involves computation of 3 velocity components in 2

space directions, has been variously termed "2.5D flow", or "slender turbulence theory",

or "slender eddy theory."

Since Pulliam and Steger used the thin-layer form of the Navier-Stokes equations, we

write, in order to retain a tridiagonal structure,

"'-_.~~a AC, P__o = A,.q 4Oq

(3).,a( - G) O =" Oq -Oq -B. + B.

aq

where the matrices A. and Bo contain no cross derivative terms, while A. and B_. contain

all such terms. The detailed structure of these matrices is given in Appendix I. Our basic

factored algorithm becomes

n(2(I-I2 2 _,G + G] + OAAq"V -h69A0 -hbj Bt hm 41

(4)2 An+2 I'M+ M,6 Bnq ±~

7

...............................-.v - .---. -- .... ...... ....... ..... ............,. . . . ..

rv.r-- M. T r r- -- .

:" 4.

This differs from the Pulliam-Steger algorithm by the addition of the last four terms on

the right hand side. Of these, the last one corresponds to a three-point backward Euler

difference scheme for time derivatives (added for accuracy and stability reasons explained

later); the next to last term corresponds to body pressure gradients imposed by the large

scale eddies, while the third and fourth from last represent the additional terms required

for the full Navier-Stokes equations in place of the thin-layer equations used by Pulliam

and Steger.

BOUNDARY CONDITIONS

The primary element in formulating our computational model involves the construction

of time- and space-dependent boundary conditions that are as realistic as possible for the

fluctuating temperature 0 and the fluctuating velocity components u, v, w at the outer

edge of the viscous sublayer. Various experimental observations are used as a guide to this

construction. Computations using the time-dependent Navier-Stokes equations are then

made over a length of time long enough for the various statistical quantities, e.g., mean

values, correlation coefficients, rms fluctuating intensities, etc., to reach a periodic state

within the viscous sublayer.

Our general approach is an extension of that employed by Chapman and Kuhn (1981)

for incompressible flow without heat transfer. They considered two coherent components

of turbulent motion at the outer edge of the viscous sublayer (lower edge of the logarithmic

region). One component represents the small scale eddies (SSE) responsible for the prin-

cipal production of turbulence and Reynolds stress. A second component represents the

organized large scale eddies (LSE) which interact with the small scale eddies. They used

a simple initial construction for purposes of illustrating the potential and main character-

istics of this type of turbulence modeling. With all quantities expressed in conventional

dimensionless wall variables, their velocity boundary conditions for the outer edge of the

viscous sublayer were

t:::!. . . . . . ... .. . . .. , .. . .. .. . ., .... , .. ... . ... .... .. . . . . . . ..."."-*."" """".,*,... .., . " ' "-,'"7"". *,-"-.,-. " " ,"," - -, .. "-". , ' ,t'-..". '. ,., -U. *-' "-. .- '. ' ., ,

Component I-SSE Component 2-LSE

u,+= 2at sin NITsin + V2(a 2 - ac)sin(N. 2T + #.2)

v,+= -2P3 sin NIT sin (5)

w,+ 2f cos NITcos + 2- ) Sine T + 0.,2)J2where a, P3, -y are the rms intensities of fluctuation of u,+, v,, w,, respectively, N, is the

*mean frequency of SSE burst events (ejection/sweep events), N, 2 is the mean frequency

of LSE, = 21rz/A is a dimensionless spanwise distance variable, T is the dimensionless

time, A is the mean spacing between high-speed or low-speed streaks, a1 /a is the uv

correlation coefficient (0.45), and ',u2 and 0,,2 are phase angles. The flow model is periodic

*..- both in time and span. Hence the boundary conditions for the spanwise side walls of the

computational domain were

U+(Y, 0,t) = UC+(Y, ,t)

+(Y,0,) = ,+( ,,t) (6)

W+(, 0,t) = We+(Y, ,t)

and at the surface

tL +(0,Z,t) = VC+(0,Zt) = wo+(0, z,t) 0

This construction of velocity boundary conditions was guided by experimental obser-

. vations of organized sublayer structure. For example, the sin " factor for u,+ corresponds

to the observation of high- and low-speed streaks spaced spanwise a mean distance of

%9

"0 _" : '.. : ". . . i '" : ,,. ' :, , " " ' ' '" - , ' ' " , " • " - ' ' " " " " " "- , . . . ' " " " " " " " " - ' " "• ° , , " " ' ' ,

. -, . ..- -% .. . .• . . . . ... •. -, .. -. .'. -.. .. .. • " -

A apart, while the factors sin f, sin ', cosq in the u,+, v,+, w,+ equations, respectively,

correspond to a simple modeling of the observations of contra-rotating vortical motion

near a wall. The 180 deg phase difference between u,+ and v,+ corresponds to observations

from conditional samples in the sublayer that u and v are 180 deg out of phase during the

-; . Reynolds-stress intensive ejection-sweep event. The 90deg phase difference in time be-

tween ve and w, corresponds to the observation that the derivative (av2/Cy+) at the outer

edge of the viscous sublayer is zero. A comparison of a number of turbulence characteristics

computed from this simple model with those measured in experiments showed surprisingly

good agreement. This model was developed for flows with a mean streamwise pressure

gradient, although numerical computations were made by Chapman and Kuhn only for

the relatively small pressure gradients that exist in incompressible pipe and channel flow.

The above two-component velocity boundary conditions at the outer edge of the vis-

cous sublayer are fully compatible with the concept of "active" and "inactive" components

of turbulent motion in the log region as characterized by Townsend (1961) and Bradshaw

(1967). Component 1 is active, involving small scale eddies, producing the Reynolds stress,

being rotational, and being dependent on wall variables. Component 2 is inactive, involv-

*ing large-scale eddies, producing energy but no Reynolds stress, being irrotational, and

being dependent on outer variables. There is much experimental information that turbu-

lent flow in the log region, and hence on the outer edge of the viscous sublayer, comprises

these two distinct types of motion.

The computational model of Chapman and Kuhn is not regarded as sufficiently real-

istic for purposes such as outlined in the section above on motivations. Their results do

". illustrate, however, that the general approach employed has the potential of eventually

leading to a sufficiently realistic model. Towards this end, it is believed that additional

aspects of turbulence physics will have to be introduced into the model. For example,

. production of Reynolds stress should be intermittent in time, rather than sinusoidal-like

as in boundary conditions (5). Also, conditional samples for v(t) in ejection/sweep events

10

...'.,. ... - .. ..o . '.-... ,*,.. . .% .: . . .- -.... . ... ', .'° . ". . ... . . ..

indicate that the v2 turbulent energy contained in these SSE events should be only a frac-

..- .. tion of the total v2 energy, rather than the entire amount as in equation (5). Further,

visualizations of viscous sublayer flow exhibit a mixture of order and disorder, rather than

-. .. the coherent order only of equation (5). For the heat transfer case we must construct

appropriate correspondingly boundary condition for the fluctuating temperature 0. at the

outer edge of the viscous sublayer.

One important guide to the construction of boundary conditions is provided by mea-

surements of spectral density f. Fortunately, spectra for 0, u, v, and w, have been obtained

by Fulachier (1972) at y-positions near the outer edge of the viscous sublayer. Data for

y+ ; 40 (interpolated between his measurements at y+ 31 and 77) are shown in figure

2. Since

j fdk j kfd(In k)-1

the areas under curves of u2kff, v 2kfv, and w2kf, versus log k are proportional to the

relative amounts of kinetic energy in these velocity components. Along the wave number

scale k, pips are shown corresponding to LSE eddy scales of 7r/6 (where 6 is the boundary

layer thickness) and to SSE eddy scales of ir/A. It is seen that, whereas both u and 0

exhibit eddy scales ranging from large [0(6)] to small [O(A)], v exhibit scales ranging only

from medium [0 (several A)] to small. Our boundary conditions will be constructed to be

as consistent as possible with experimental data of this type.

A relatively simple method (developed under a separate contract) is used to simulate

the intermittent character of Reynolds-stress production. Since UV is produced only by

the active component of small scale eddies (SSE), only the time functions sin N1T and

cos NIT in equation (1) need be modified. For this purpose we use a simple Fourier series

approximation I(0) to a rectangular pulse function, where NIT.

*$.*r.-'-11

• ~~...-............ ... ........... . . . .....--...... •.... . •... -...... ,-%.,,,.-°,.-.,

For M terms in the approximating Fourier eries, we have

Af

H(Ob) = (an cos no6 + bn sin no)

2 (7)

an [sin nX, - sin nX2]

=n -I- [X1 (cos nX2 - 1) + (cos nX1 - 1)]

values of M in the range of 3 to 5 are anticipated. Initial computations will be made with

X,= X2= 0.25ff, since hot-wire measuirements have indicated that the principal uV is

produced in about 25% of the time, with relative uv quiescence the remaining 75% of time.

r-- ,(o, 1)

X1

12

. A .,..... .. ... .*

In conventional wall variables, the outer edge boundary conditions constructed to date

include terms representing small scale eddies (SSE) of length scale A, large scale eddies

(LSE) of length scale > 10A, and medium scale eddies (MSE) of intermediate length scale

3A. Computations are made within a domain covering 3 spanwise cells each A in width.

The LSE are treated as being of sufficiently large scale that their time-dependent velocity

component does not vary across the spanwise extent of the computational domain. The

equation of continuity precludes having a LSE component in the boundary condition for

Vt.

SSE LSE MSE

Scale A Scale >1OA Scale 3A

t= V204, F,() sin + \2( - ) sin (N 2T + 002)

Vt= V2P, F.(0) sin S + 2/32 sin (3N,.2T) sin3

wt =v'if 1F~(~)cs~ ± 2(9 23f sin (UT +02)

(8)

0.= 5a FO)sin + VP2(a- a')sin(N.2T + €02) (9)

where, in addition to the quantities already defined, a =7 \/ , €02 is a phase angle in

the large-eddy component of temperature, and

H(O)

(10)H(O + Owl)

13

•-..

-. '".

P, . V,.4 I4 P- IV _.Y 1 -. XJ . 7. . . .. . 1 " .- 1--.- 1 " 7- .

are the intermittent time functions normalized such that P = 1. The boundary conditions

for the spanwise side walls of the computational domain are

ue+(y,o,t) = u.+(y,3A,t)

V-+(y,0,t) = v.+(y,3A, t)

S0, t) = w+(y,,t)

Thus the computation allows for flow interaction between the center cell and the two

outside cells unconstrained by specific boundary conditions.

As in the model of Chapman and Kuhn: a, 1, -, a are determined from experimental

'. data on the intensities of turbulence fluctuations in wall units at the outer edge of the

viscous sublayer (a -, 2,3 1, - 1.3, a - 1.3); O2, '002, 0.2, are determined by

--- computer trial to conform as well as possible to the mean-velocity law of the wall, mean-

temperature law of the wall, and u-velocity skewness, respectively; 0' is determined

analytically by the requirement that (aP/O0y), is zero; a, is determined by the correlation

coefficient 0.45 = - = a1 13/afl; a, is determined by the correlation coefficient

0.8 -(Ruo)e, = + V -a)(a - a)cos (002 - 0.,2); and 1/1, is determinedlita

from conditional samples of the ejection/sweep event. This latter determination is made

by assuming that the peak to peak amplitude ratio < v >,p / < u >pp,, from the

conditional samples is equal to the ratio flt/a 1, and then using the above equation for

-.. (R,.) to compute both 1l/3 and a1 /a. An independent determination of a1 /a has been

made from direct measurements of the fraction of total u' energy that exists during bursts,

and hence also of 1/P3 from the equation for (R,,,),. Still a different determination has

been made from < w >, / < u >pp= Pl3/a 1 coupled with the equation for (R,,),.

14

--+. * rz r+r . . s v .- ' -.

Results of these determinations (made on a separate contract) are:

Data Source Method PI/PChen & Balckwelder (1978) <v> / <u> plus R = -. 45 .53Nakagawa & Nezu (1981) < v > /< u > plus R. = -. 45 .72Blackwelder & Kaplan (1979) < v > / < u > plus R., = -. 45 .49Kim (1982) < u > / < v > from LES computations, .60

plusX4,V = -. 45Kim, Kline & Reynolds Fraction of total u2 energy during .59(1971) bursts (.68), plus R,, = -. 45Blackwelder & Kaplan (1979) < w > /< u > plus R, = -. 45 .64

On average, the value of p1/,8 appears to be near 0.6.

It is to be noted that the structure of boundary conditions (8) comprises in

equal proportions both ejection/sweep events (where a sweep follows an ejection), and

sweep/ejection events (where an ejection follows a sweep). The recent experiments of

Johansson and Alfredsson (1982) clearly reveal that both types of event occur. Their ex-

perimental VITA data for the smallest threshold values and the longest integration times

employed, indicate these two event types to be about equally numerous. Both types have

also been observed visually by Offen and Kline (1974) who use the term "cleansing sweep"

in an ejection/sweep event to distinguish from sweeps in a sweep/ejection event.

15

., ,. . . .. . . . .

STATUS OF RESEACH

I. Recapitulation of Research During 1982

During the first year of research, effort was directed mainly on surveying various ex-

isting theories for Pri in the viscous sublayer, on programming a Navier-Stokes code, and

on testing and debugging this code.

Survey of various theories for Prt

Without a reliable guide from the experiment as to how Prt should be modeled near

a wall, the various theoretical models advanced to date have differed greatly. This is

illustrated in figures 3, 4, and 5 showing a compilation of various theories for fluids of

molecular Prandtl number 0.72 (air), 6 (water), and 1000 (oil), respectively. These various

theories are for zero pressure gradient and clearly demonstrate that a very wide range of

uncertainty exists for Pr in the viscous sublayer. Present theories about the dependence

of Prt on pressure gradient are fewer in number but also differ greatly.

Programming of Navier-Stokes Code

The implicit, compressible-flow, 3-D code of Pulliam and Steger (1980) has been

used as the basic Navier-Stokes code to modify and adapt for viscous sublayer modeling.

This code has been modified in five ways as follows: (1) the grid has been stretched to

cluster points near the wall and the outer edge; (2) the flow variables have been made

nondimensional in the conventional wall variables used for turbulent flows; (3) various

necessary viscous terms have been added to bring the Pulliam-Steger "thin-layer" code

into the full Navier-Stokes form; (4) the boundary conditions have been restructured to be

appropriate for computations of viscous sublayer turbulence; (5) the algorithm for time

derivatives has been revised for increased accuracy.

Testing and Debugging of Code

In order to test the algorithm accuracy for the time derivative terms, an exact an-

alytical solution was derived for incompressible oscillating shear flow with heat transfer.

This solution is an extension of the Chapman-Kuhn solution for flow without heat trans-

16

- ]

J.

2.." 1 . '. ° _. r. --r_ W ,,. , ,L ' - .,K *,5 -V , -7~o w7 , ; ° % . . . .,7: , oO. . . . . '

fer. Mathematical details of the analytical solution are presented in Appendix H. It was

' found that neither of the option algorithms in the original Pulliam-Steger code (i.e., Euler

" -implicit first order, or trapezoidal second order) were sufficiently accurate for turbulenceC- C

'':. " computations. With helpful guidance from Dr. Pulliam, a three-point backward Euler

"-. C- algorithm was coded and found to be satisfactory. The very close agreement obtained

*I with this second-order accurate algorithm, between numerical computations and the ex-

act analytical solution, is illustrated in figures 6 and 7 for the adiabatic-wall case and

. the heat-transfer case, respectively. Since oscillating shear flow with heat transfer involves

viscous and heat-conduction terms as well as time-derivative terms, this test flow provides

a check on the accuracy of the numeries for all three types of terms. It does not provide,

however, a check on the accuracy of the important non-linear convection terms which are

zero in oscillating shear flow.

Our initial runs of the viscous sublayer code revealed an instability problem associated

with a large residual in the continuity equation, and a "decoupling" of the computed data

in the y-direction normal to the wall. In this decoupling, computed values of v),p and p'

oscillated from grid point to grid point. Decoupling is usually a signal of a problem with

boundary conditions. (This instability was minimized to acceptable levels by a coding

change described below in the section on progress during 1983.) Examination of the

continuity residual revealed that the instability arose at the outer edge of the viscous

sublayer, since there the residual was around 10 - , whereas in the inner main portion of the

flow it was the order of 10- to 10- . This also suggests that the initial boundary conditions

. at the outer edge were creating a problem. Such problems can arise form either physical or- numerical causes. Some trials with obviously wrong boundary conditions produced very

extreme decoupling, leading to a blow up of the code.

"" II. Research Progress During 1983

O. During the past year, research has been focused on completing the programming, on

investigating various types of boundary conditions, and on conducting some preliminary

17

• .'.. "C . . .

• '. '" " " -. . . . • " • ." , '. " " -.. ', " .' .-.***'*. '', e . . . ,' " - . '-*' , -' ' . ' - " _ -

% r~

computer runs to explore the effects on turbulent Prandtl number of variations in molecular

Prandtl number and in mean streamwise pressure gradient.

Programming Modifications

Considerable code modification was undertaken in order to expand the data output

and graphics program, to increase code robustness, and to transfer the code to two new

computers installed at the Ames Research Center during 1983. The data output and associ-

ated graphics program was extended to handle 26 statistical quantities: the mean velocities

U, T), 11, temperature 0, pressure p and density , the Reynolds stresses try, Tow and velocity-

. temperature correlations u0, vO, the intensities of fluctuations u', VI, WI, 0', P' ', (uv)',

the correlation coefficients R,,, RO0, R 0, the skewness S,,, So, S, and the flatness fac-

tors F, F9 , F,. It was found that refinement of the grid from 25 x17 to 39 x 25 in

the spanwise plane increased considerably the codes' robustness in handling a variety of

boundary conditions that were explored. The finer grid apparently localizes boundary-

condition problems near the outer edge of the viscous sublayer, and allows for a greater

portion of the interior flow to be well behaved. Some tests were made of adding an artificial

dissipation subroutine to the code in order to assess its effects on the stability and decou-

pling problems. Both stability and decoupling were improved, but the numerical results

were altered considerably, and hence the artificial dissipation subroutine was dropped from

further consideration.

During 1983 NASA Ames replaced the CDC 7600 computer, which we had originally

programmed for, first by the CRAY--IS, and then later replaced this computer by the

CRAY-XMP. The corresponding changes in operating system required some modifications

"- to the code. To date all three of these computers have been used in the present research

project.

":- .Research on Boundary Conditions

* In a code for 3-D compressible-viscous flow, five boundary conditions are required

at every boundary. Not any combination of five conditions will work. This problem is

18

..

particularly acute at the out edge of our computational domain where non-zero velocity

components are specified and where fluid flows in and out across this boundary. The set

of boundary conditions which is physically and mathematically correct for this type of vis-

cous compressible flow is not known a priori. Since no sound theoretical guide is available,

considerable computer trial is necessary using various types of boundary conditions. Fur-

ther complexity arises because the numerical boundary conditions can be treated either

explicitly or implicitly. To date explicit boundary condition schemes have been mainly

used. One implicit scheme has been tried, but has not provided an improvement in either

accuracy or stability.

In the present research we have generalized the boundary conditions of Chapman

and Kuhn (1981) who worked with incompressible flow. Difficult problems of boundary

conditions arise when it is attempted to compute a compressible flow. If the dissipative

terms were to dominate at the outer boundary, then a stable scheme would likely occur

-. -with the specification of velocities u., v, W., temperature 0,, and either pressure p, or

*.k density p,. If the non-linear convective terms dominate, however, then this may only be

weakly stable at best. Under such circumstances a characteristics type set of boundary

conditions might have to be implemented. With Navier-Stokes codes the situation is not

usually near either of these two extremes, but is somewhere in between. Specification of

wrong boundary conditions can result in serious stability problems. In the present research

we have attempted to structure boundary conditions that use as much as possible of the

previous work of Chapman and Kuhn, and still provide adequate stability.

One change was found to be necessary to achieve stability. At the outer edge, the

O: mass flux quantities (pu),, (pw)e, (pw), are specified instead of the velocity components.

In this way mass is conserved within the computational domain. For the fourth boundary

condition the outer-edge temperature O, has been modeled in a structural form similar to

0. that of the u-component of velocity. For the fifth boundary condition at the outer edge of

the viscous sublayer, there are a number of possibilities of which the following have been

. .. . . .. .. . .

investigated:

(a) p, extrapolated from y-momentum equation

(b) p, extrapolated from continuity equation

(c) p. = [. 1 + esin(NIT + 01) + b sin(NIT + 02)]

(d) p, = [1 + sin(NIT + 01) + Ssin(N, T+ 02)1-'

* (e) p. extrapolated from (), = 0

It is noted that boundary conditions of the types (a) and (b) do not introduce any addi-

tional physical assumptions about the turbulence, and hence are the most desirable. Thus

far, the most success has been obtained with conditions (a), and the numerical results

- presented later were obtained using this condition.

Two quite different types of boundary condition also have been explored wherein

derivatives of velocity are modeled:

(f) (pv),, (pw)., o specified with condition (a) and ( =( )1+esin(NiT±# sinl

6 sin(N1 T + 02)]

(g) (pu),, (pv),, (pw)e specified with condition (a) and ) ( +) [1i ± Esin(NIT +

-1) sin t + 6 sin(N IT + 02)]

These two types of boundary conditions yielded somewhat more realistic values for the

temperature intensity 0' and the fluctuating pressure intensity p'. Other quantities such

as mean velocity Fi and mean temperature 9, however, were less realistic.

One further generalization in the structure of the outer-edge boundary conditions was

coded in 1983. The Fourier series equations representing intermittent Reynolds stress pro-

duction, equations (7) and (10), have been included in the computer program. Exploratory

cases with this type of boundary condition will be run during 1984.

Preliminary Results for Various Pressure Gradients and Molecular Prandtl Number

Our computational results to date have been obtained with boundary conditions on

O (pu/)., (pv/l),, (pw/ p) specified to be the same as Chapman-Kuhn (i.e., the same as the

right hand side of equations (5)), with condition (a) above imposed, and with O specified

20

as

.- 2a, sin(NIT + ,)e) sin C + 42(a2 - a') sin(N.2T + 4)92)

Here a is the intensity of temperature fluctuation at the outer edge (a = 1.5 in wall.c. -'

variables). The phase angles 400, 4'2, and the coefficient a, for these preliminary results

were selected so that the u9 correlation coefficient would be -0.8 at the outer edge and

the vtO correlation coefficient 0.45. These preliminary results serve to illustrate the type ofresults to be produced by our Navier-Stokes code, as well as the qualitative dependence

of Pr on the parameters such as pressure gradient and molecular Pr.

In figure 8 a comparison is presented of our preliminary computations with experimen-

. tal results for mean temperature 0, intensity of temperature fluctuation 0', and the two

temperature-velocity correlation coefficients PX9 and R#. These data are for air, Pr = 0.7,

with zero pressure gradient and an isothermal wall. The computed mean temperature field

and &# agree well with experiment, but the computed temperature intensity and /,.e are

higher than experiment. Computed values of R,.9 are believed to be high because all of the

modeled v energy in the boundary conditions of equation (5) is contained in the small scale

eddy component which is correlated with the small scale eddy component of temperature.

It is anticipated that use of a more realistic boundary condition for v,, such as represented

.3- by equation (8), wherein only part of the v energy is contained in the correlated small

S. scale component, will lower the computed R,,9 to values in closer agreement with experi-

ment. Reasons for the relatively high values computed for temperature intensity are not

understood at the present time. Further research with varied boundary conditions should

clarify this.

Some computations have been made in order to provide a preliminary assessment of the

effects of molecular Prandtl number Pr on the turbulent Prandtl number Pr in the viscous

• "sublayer. Runs were conducted for Pr = 0.7, 1.5, 3, 6. Higher values of Pr encountered a

stability problem which is being investigated. The computed results for an isothermal wall

21-.:..-5.-

are presented in figure 9. In some limited respects these results agree qualitatively with

two of the various theories advanced to date, namely, the theories of Cebeci (1970) and

Geshev (1978). The summary of various theories in figures 3, 4, and 5 include these two.

It is interesting that the computed effect of Pr on Prt near the wall is opposite to that

near the outer edge of the viscous sublayer. The computed curves intersect near y+ = 5.

Near the wall, our computed Pr increases considerably with Pr. Geshev's theory for an

isothermal wall indicated a similar increase.

Values of Prt at y+ 2

PresentPr Computation Geshev Cebeci0.7 .98 .85 1.256.0 2.0 2.2 1.04

Other theories among those compiled in figures 3-5, however, do not indicate such a

strong increase of Prt with increasing Pr near a wall; and Cebeci's theory indicates that

Pri near the wall decreases with increasing Pr. It is noted that Geshev's treatment of

wall temperature boundary condition is more realistic than the others, and affects 0- and

hence Prt near the wall in an important way. Near the outer edge of the viscous sublayer

the effect of Pr on Prt is relatively small, as might be anticipated. The computed trend

of Pr > 1 for Pr < 1 and Pr < 1 for Pr > 1 is consistent with experimental data. In

this case Cebeci's theory indicates the same trend but Geshev's does not.

Values of Pre at Y+ = 20

PresentPr Computation Geshev Cebeci

0.7 1.12 .95 1.106.0 .85 1.10 1.00

The various theories do not exhibit the near independence of Pri on Pr around y+ - 5K.:- that our preliminary computations exhibit.

Some preliminary computer runs were also made to explore the effects of mean stream-

wise pressure gradient on Pr for air (Pr = 0.7) flowing over an isothermal wall. In this

22

case the boundary conditions were the same as above, except that the bursting frequency

N, of small scale eddies was changed with pressure gradient in accordance with the ex-

perimental data of Schraub and Kline (1965). The values selected for pressure gradient

in wall variables were dp+/dx+ = -. 02 (favorable), 0, +.02 (adverse). As illustrated in°' 4

figure 10, the computed results indicate relatively little effect of mean streamwise pressure

gradient on Prt. In this case also the qualitative effect of dp/dx on Pr near the wall is

opposite to that near the outer edge. Near the outer edge, the trend of increasing Pre

with increasing favorable pressure gradient (decreasing dp/dx ) is qualitatively consistent

with the trend deduced from experimental data (Blackwell (1972), Gibson (1981)). This

trend is the opposite from that indicated by the recent theory of Thomas (1982). Below

is a comparison of Pre values near the wall.

Values of Pr at y+ = 5

dp Present Thomas Blackwell0 d+ Comiputation Theory Experiment-.02 .950 1.12 .85 1.80+.02 1.12 1.00 1.60

Near the outer edge of the viscous sublayer the corresponding results are:

Values of Pr at y+ = 20

,dpi Present Thomas BlackwellComputation Theory Experiment

-. 02 1.20. 0 1.12 .90 .85

+.02 1.00 1.10 .75

Unlike our computed results, neither the results of Thomas or of Blackwell exhibit the

crossover and nearly common intersection near y+ ' 12.5 of the curves for various dP4 Idx+.

It is emphasized that, since our present quantitative results are preliminary and will

* .likely change somewhat as the temperature and velocity boundary conditions are refined,

the precise numerical values are not regarded as significant. The observed qualitative

23

.. . . . . . .

... ..

trends, however, such as the strong increase near the wall of Pro with increasing Pr, and

the relatively small effect of dp/dx on Pr, may not be markedly affected by such changes

in these boundary conditions. These preliminary results serve to reveal mainly the types

of information produced by the Navier-Stokes code being developed, and the qualitative

trends which may not be altered as boundary conditions on temperature and velocity are

further refined. If the preliminary result of only small effects of dp/dx on Pr holds up

with more refined boundary conditions, this would be a simplifying and helpful result for

practical computations of heat transfer. These preliminary results serve to illustrate the

inconsistencies and deficiencies of the various theoretical models for Pr in the viscous

sublayer.

24

... .

.. . .. .... ...... .-.-.-. ...... ,..............-.-'.-.--,..- .,.., .-... -'.

I11. Outline of Research Anticipated During 1984

Details of the research effort planned during 1984 have been given in our recent pro-

posal dated July 1983 for continuation of the present research project. Very briefly, the

main effort will be expended in the areas of: (1) refining the temperature boundary con-

dition; (2) using more complex velocity boundary conditions such as those representing

intermittent Reynolds stress productions; and those involving two components of v veloc-

ity; and (3), to the extent that time permits, investigating ways of further reducing the

n decoupling observed in f3, p, and p' very near the wall, and ways of further improving the

stability and robustness of the code.

A.2

° .

_-.., -

9 -

, -. -. . - ... * -- -

REFERENCES

[1] Back, L. H., Cuffel, R. F., and Massier, P. F. (1970): Effect of Wall Cooling on theMean Structure of a Turbulent Boundary Layer in Low-Speed Gas Flow. Int. J. HeatMass Transfer, 13, pp. 1029-1047.

[21 Beam, R. M. and Warming, R. F. (1978): An Implicit Factored Scheme for the Com-pressible Navier-Stokes Equations. AIAA Journal, 16, pp. 393-402.

[3] Blackwell, B. F. (1972): The Turbulent Boundary Layer on a Porous Plate: AnExperimental Study of the Heat Transfer Behavior with Adverse Pressure Gradients.Report HMT-16, Dept. of Mech. Eng., Stanford University.

[41 Blackwelder, R. F. and Kaplan, R. E. (1976): On the Wall Structure of the TurbulentBoundary Layer. Jour., Fluid Mech., 76, Pt. 1, pp 89-112.

. [5] Bradshaw, P. (1967): "Inactive" Motions and Pressure Fluctuations in TurbulentBoundary Layers. Jour. Fluid Mech., 30, Pt. 2, pp 241-258.

[6] Cantwell, B.J. (1981): Organized Motion in Turbulent Flow. Ann. Rev. Fluid Mech.,13, pp 457-515.

[7] Cebeci, T. (1970): A Model for Eddy Conductivity and Turbulent Prandtl Number.Rept. No. MDC-J0747101, Douglas Aircraft Co., Long Beach, California.

[8] Chapman, D.R. (1980): Trends and Pacing Items in Computational Aerodynamics.Lecture Notes in Physics, Springer-Verlag, 141, pp. 1-11.

" [9] Chapman, D.R. and Kuhn, G.D. (1981): Two-Component Navier-Stokes Computa-tional Model of Viscous Sublayer Turbulence. AIAA paper 81-1024, Proc. 5th CFDConference, Palo Alto, California.

[10] Chen, C.H. and Blackwelder, R. (1978): Large-Scale Motion in a Turbulent Boundarylayer: A Study using Temperature Contamination. Jour. Fluid Mech., 89, Pt. 1, pp.

• " 1-31.

[11] Clark, R.A., Ferziger, J.H. and Reynolds, W.C. (1979): Evaluation of Subgrid-ScaleModels Using Accurately Simulated Turbulent Flow. Jour. Fluid Mech., 91 Pt. 1,pp. 1-16.

112] Fulachier, L. (1972): Contribution a L'Etude des Analogies des Champs Dynamiqueet Thermique dans une Couche Limite Turbulent. Effet de L'Aspiration. Thesis,University of Provence, France.

[13] Elena, M. (1977): Etude Experimentale de la Turbulence Au Voisinage de la Paroi. D'un Tube Legerement Chauffe. Int. J. Heat and Mass Transfer, 2_0, pp. 935-944.

[14] Geshev, P. I. (1978): Influence of Heat Conduction of the Wall on the Turbulent

26

",',,,. .. '°...'...'.....'.. "............................ .... """ ... "" "'" "" .... "

-~~~-C 7 -1 -7 7 .-. -. ; .. ... -~ - .k I. 'W. -. '. - -.*- .* . . .

Prandtl Number in the Viscous Sublayer. Inzhenerno- Fizicheskii Zhurnal, 35, pp.292-296.

[15] Gibson, M. M., Verriopoulos, C. A. and Nagaro, Y. (1981): Measurements in theHeated Turbulent Boundary Layer on a Mildly Curved Convex Surface. Turbulent

• -Shear FLow, Third International Symposium, pp. 80-89.

[16] Hishida, M. and Nagaro, Y. (1979): Structure of Turbulent Velocity and TemperatureFluctuations in Fully Developed Pipe Flow. Journal of Heat Transfer, 101, pp. 15-22.

[17] Iritani, Y., Kasagi, N. and Hirata, H. (1981): Transport Mechanism in a TurbulentBoundary Layer-Visualized Behavior or Wall Temperature by Liquid Crystal. Sub-mitted to Trans. JSME (B).

[18] Jischa, M., and Rieke, H. B. (1979): About the Prediction of Turbulent Prandtland Schmidt Numbers From Modeled Transport Equations. Int. J. Heat and MassTransfer, 22, pp. 1547-1555.

[19] Johansson, A.V. and Alfredsson, P.H. (1982): On the Structure of Turbulent ChannelFlow. Jour. Fluid Mech., 122, pp. 295-314.

[20] Johnk, R. E. and Hanratty, T. J. (1982): Temperature Profiles for Turbulent FLowof Air in a Pipe. Chem. Eng. Sci., 7, pp. 867-881.

[21] Kader, B. A. (1971): Temperature and Concentration Profiles in Fully TurbulentBoundary Layers. Int. J. Heat and Mass Transfer, 24, pp. 1541-1544.

[22] Kays, W. M. and Moffat, R. J. (1975): The Behavior of Transpired Turbulent Bound-ary Layers. Studies in Convection, Vol. 1, pp. 223-319.

[23] Kim, J. (1982): On the structure of Wall Bounded Turbulent Flows. Paper presented

at Amer. Phys. Soc. Division of Fluid Dynamics Meeting, Rutgers Univ., Nov. 21-23,1982.

[24] Kim, H. T., Kline, S. J. and Reynolds, W. C. (1971): The Production of TurbulenceNear a Smooth Wall in a Turbulent Boundary Layer. Jour. Fluid Mech., 50, Pt. 1,pp. 133-160.

[25] McMillan, 0. J. and Ferziger, J. H. (1979): Direct Testing of Subgrid-Scale Models.AIAA Paper 79-0072.

[26] Meek, R. L. and Baer, A. D. (1973): Turbulent Heat Transfer and the Periodic Viscous

Sublayer. Int. J. Heat and Mass Transfer, 16, pp. 1385-1396.[271 Nakagawa, H. and Nezu, 1. (1981): Structure of Space-time Correlations of Burting

Phenomena in an Open Channel Flow, Jour. Fluid Mech., 104, pp. 1-43.

27

#... .. . .. . . ...... . . . . . . . . . . .

1281 Offen, G. R. and Kline, S. J. (1974): Combined Dye-Streak and Hydrogen-BubbleVisual Observations of a Turbulent Boundary Layer. Jour. Fluid Mech., 62, Pt. 2,pp. 223-239.

1291 Patel, V. C., Rodi, W. and Scheuerer, G. (1981): Evaluation of Turbulence Models forNear-Wall and Low-Reynolds Number Flows. Proc. Third Symposium on TurbulentShear Flows, Sept. 9-11, 1981, pp. 1.1-1.8.

1301 Pulliam, T. H. and Steger, J. L. (1980): Implicit Finite Difference Simulations ofThree-Dimensional Compressible Flow. AIAA Jour., 8, pp. 149-158.

[31] Reynolds, A. J. (1975): The Prediction of Turbulent Prandtl and Schmidt Numbers.Int. J. Heat and Mass Transfer, 18, pp. 1055-1069.

[32] Rogallo, R. (1977): An ILLIAC Program for the Numerical Simulation of Homoge-. neous Incompressible Turbulence. NASA TM NO. 73, 203.

[33] Schraub, F. A. and Kline, S. J. (1965): A Study of the Structure of the TurbulentBoundary Layer with and without Longitudinal Pressure Gradients. ThermosciencesDiv. Report MD-12, Stanford University.

[34] Shirani, E., Ferziger, J. H. and Reynolds, W. C. (1981): Mixing of a Passive Scalarin Isotropic and Sheared Turbulence. Report No. TF-15, Thermosciences Division,

- . Dept. Mechanical Engineering, Stanford University.

[35] Simpson, R. L., Whitten, D G. and Moffat, R. J.: An Experimental Study of theTurbulent Prandtl Number of Air with Injection and Suction. Int. J. Heat and MassTransfer, 13, pp. 125-143.

[36] Snijders, A. L., Koppius, A. M. and Nieuwvelt, C. (1983): An Experimental Determi-nation of the Turbulent Prandtl Number in the Inner Boundary Layer for Air Flowover a Flat Plat. Int. J. Heat Mass Transfer, 26, pp. 425-431.

[37] Subramanian, C. S. and Antonia, R. A. (1981): Effect of Reynolds Number on aSlightly Heated Turbulent Boundary Layer. Int. J. Heat and Mass Transfer, 24, pp.1833-1846.

[381 Thomas, L. C. (1982): A Turbulent Burst Model for Energy Transfer in the WallRegion. Heat Transfer 1982 Proc. of the Seventh Int. Heat Transfer Conf., Vol. 3,pp. 307-312.

[39] Thomas, L. C. and Benton, D. J. (1982): A Turbulent Burst Model for BoundaryLayer Flows with Pressure Gradients. Heat Transfer 1982 Proc. of the Seventh Int.Heit Transfer Conf., Vol. 3, pp. 313-318.

[40] Townsend, A. A. (1961): Equilibrium Layers and Wall Turbulence. Jour. Fluid Mech.,11, pp. 97-120.

28

.. . .,.,.. . ., . . ... .... , v .. . . , ,,., .. ". . ' , -". . . .. ., .. , .. . ., , . . j' , .. ..,, , . . . . .... , . . ... .

-.1--rIJ

APPENDIX I

"" "~ DECOMPOSITION MATRICES

The derivatives 0F/8q and a /Oq are each decomposed into two matrices, one (subscript zero)

that does not contain cross derivative terms, and another (subscript z) that contains all such terms.

.-.?:A, o + A,

=B. + B

...-.. Oq

where the matrix A. is

0 0 17Y 0 0-U1lyVt Y 0 0

"W' ,€ - V') -17,(-y - 1)U -tlv("y- 1) -17.,(,y - I) W I(qY- 1)

"- .A 0 - w ryly 0 nw 170 0

7.7v [2 ] - yfl - 1)ulv -(- - 1)v --( 1- )w)1v Vyv-[ e y ]

*.vo

0 0 0 0 0-8,6(u/p) 8,(5 ) 0 0 0

-s 26,,/oP) 0 S26 ) 0 0

..-...4.. +€S-16s ((- 1)(

-....: 29

,0 - -. -- : - - " - ' " ,,., .,,;,,<:.:::: ..,...........,........-., .... .. .....

i .- . . . - - - -

.1'.

2 1-)(U 2 + VI+W2)* . N.

?-

2 4S1 L7V 82=-Si 85=S

3 P

J = Jacobian - 1

and the matrix A, is"4

0 0 0 0 0

0 0 0 0 0

-S45. (!) o 0 8,6,( ) 0

A 2 --16 0

0 0 s46fW+S 3WS S06C (') + 0S6CV 0

S3 = I111~z

284= -- 833

The matrix B,, is

w0 0 0*!; -.- uw ~ .w 0 u0

Bo-- -V ,w 0 0 +w2~ W2 - 1)u - Iy )v -2)w 1

2 W [22 - 4 Q )U .W -e )V ,W 02 - -('-1] -yW

30

. . .% * . * * * * -*. - - * * * -.

-T6b ( 0 T16C Op) 0 0-T2 6 (l 0 0 T2 6b(I 0

-T6 p°"o

J (7'S - To)b(!!!! (TI TS)5b( (TI T5)6C (E) (T2 -T 5)5C (M) T56C(~

1+(T 5 - TIO0 TO6

T.. -T T 4) 2 T -IT.

3 P 1

and the matrix B. is

0 0 0 0 0

0 0 0 0 0- 6 q () 0 0 ( T (o -)t 0

Bi " -4 0 846,0 0 0

83 [!,,W + vj- , (,) 0 84W6 1, (,) + 8 3' ( p) 0

-84 [!Pv" + W6 .1 / }.- L1u I31

0t.

0...

.:-.7.1

Q-"

APPENDIX II

An exact analytical solution has been derived for oscillating shear flow with heat trans-

fer in an incompressible fluid with constant properties. Since the momentum equations

are independent of the thermal energy equation in incompressible flow, the velocity field

is the same as for the case without heat transfer derived by Chapman and Kuhn (1981).

With y as the coordinate normal to the infinite flat surface, and u and v as the velocity

components in the streamwise and spanwise directions, respectively, we have

u = A [sinn t- e-knv sin(nt - kny)] + cy

w = B Isin(mt + ( - ey sin(mt - kmy)]

where

The oscillating frequencies are n in the streamwise direction and m in the spanwise direc-

tion, and the mean velocity profile is U = cy. For this velocity field, the solution to the

thermal energy equation

80 a2 (,U2 / \V21pc' i=k -+p ROy + y

is

T(z, t) By 2 + C1Y + C2 + B, 2knv + B 3 e - 2kmv

+ A 2e- * "y [sin(nt - kay) - cos(nt - kny)] - A~e- 2kn cos 2(nt - k.y)

0- A4e- 2 km cos 2(mt + k - k,,y) - A 2e- 0" Vsin(nt - 81 y)

2- Ae-' cos(nt - , 1y) + A3e-2V cos(2nt - ,6)

0

+ A4e- Oay cos(2mt + 2# - f3y)

32

-"9 . . , . . . . . . , , .• , , , , , . . .. .: ..• • : . : :' . ... : , - " " . .. . . . . . . . . - . ., - - ., : .. , . - - .. - , , - - . , . . .: .

fhi = Prt 62 nPr? 83 mPrt

B1 -__ A 2 - 2Ack,,B C, C~Re(n 2k-#)

23 2 pr A3 140PA - CORe (2n - _

B3 _.2AP+ B2 24 k

4 pCpe(2

S.p

20,

0.-

33

2.5

2.0

1.5

Pr

* 1.0

.1 ~.5

viscous sublayerI

01 10 Y 0

Figure 1. -Envelope of experimental uncertainty for turbulent Prandtl numberfrom data of Simpson et. al. (1970).

.2

(-u2/q2)k- q2 2 + V2+ 2

* 01 10 102 k,m- 13 0

Figure 2. Spectral density of the three components of velocity fluctuation at

Y+ 40 from data of Fulachier (1972).

- -. .- *. S * *ell

a,9

rN

z -z

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L A Back.Cuffel.Masster 19694- Kader 1979

Gibson,Verrlopoujous,llagano 198B0SnIjdars.Iroppjus,NfeuWvelt 1931V obhnkd Han ratty 196?k CM~ KaYsIiof fat 1975

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0 Chen,Blzack%---lder 1976

U))j

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Vj 0 0 or 9 V -b

C. 4____

0.0 10.0 20.0 30.0 40.0 0.0 10.0 20. 0 30.0 40.0Y +

4

Fi ur 8. C m a i o f p e i i a y cop t t o s ( o i i e i h e p r m n

r0u t fo mea an0l c u t n e p r t r , a d f r t e t m e a u e

velcit coreato coff3ens Moeua1r3.,zropesrgradent

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-. 0 0.

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C)

LEGENDt, PR - 0.

* * PR -6.O

0.0 2.5 5 .0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5

Figure 9.Preliminary computations of turbulent Prandtl number for

fluids of different molecular Prandtl number. Zero pressure

gradient.

TURBULENT PR WITH VRRYING DP/DX

If)

Favorable

Ci-4

CL

Adverse

LEGEND

Adp/dx - +.02

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5

Figure 10. Preliminary computations of turbulent Prandtl number fordifferent mean streamwise pressure gradients. SMolecular Pr =0.7, isothermal wall.

-.--.-.. - -. ,-: -...

Initial Stiffness CyclesStress Strain Modulus toMixture Temperature, OF psi Inches/Inch psi Failure

E 77 21.8 0.00084 25,952 4,71022.2 0.00059 37,627 7,62030.2 0.00117 25,812 2,82015.3 0.00037 41,351 78,300 .14.0 0.00027 51,852 56,25412.7 0.00020 63,500 260,27716.4 0.00030 54,667 83,42910.7 0.00069 15,507 35,13533.8 0.00162 20,864 2,71923.3 0.00097 24,021 6,550

40 129.8 0.00039 332,821 20,461120.0 0.00032 375,000 30,501164.4 0.00047 349,787 10,49397.8 0.00021 465,714 113,247106.7 0.00032 333,438 83,394 ..151.1 0.00055 274,727 8,000175.5 0.00081 216,667 1,632164.4 0.00040 411,000 5,730186.6 0.00074 252,162 2,400112.0 0.00028 400,000 109,546

F 77 21.6 0.00060 36,000 21,18032.4 0.00158 20,506 1,13316.0 0.00029 55,172 50,04814.0 0.00029 48,276 78,35018.2 0.00038 47,895 20,21912.7 0.00029 43,793 145,08020.0 0.00068 29,412 6,45524.0 0.00063 38,095 10,39726.7 0.00061 43,770 2,624

40 133.3 0.00086 155,000 2,618142.2 0.00049 290,204 9,411164.4 0.00063 260,952 3,741175.5 0.00073 240,411 1,591111.1 0.00047 236,383 8,40097.8 0.00024 407,500 95,949

122.2 0.00043 284,186 10,046186.6 0.00069 270,435 1,680120.0 0.00052 230,769 7,491102.2 0.00030 340,667 37,21196.0 0.00020 480,000 149,100

A3

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. . ... . .t" 'a'a aL.~.a~

Initial Stiffness CyclesStress Strain Modulus to

Mixture Temperature, OF psi Inches/Inch psi Failure

G 77 43.3 0.00152 24,487 99834.4 0.00094 36,596 2,80822.0 0.00063 34,921 8,25017.8 0.00048 37,083 16,55830.0 0.00095 31,579 2,47216.7 0.00036 46,389 64,19819.8 0.00071 27,887 8,46014.9 0.00043 34,651 27,92813.8 0.00029 47,586 94,02415.6 0.00033 47,273 68,049

40 151.5 0.00085 178,235 675129.8 0.00067 193,731 1,093109.3 0.00042 260,238 11,680120.0 0.00048 250,000 4,24497.8 0.00039 250,769 7,26386.7 0.00035 247,714 17,46966.4 0.00030 221,333 45,16677.5 0.00028 276,786 43,46657.8 0.00023 251,304 93,982

H 77 20.0 0.00240 8,333 33010.7 0.00122 8,770 1,0658.2 0.00036 22,778 17,8626.7 0.00039 17,179 28,2905.1 0.00019 26,842 285,6025.8 0.00032 18,125 38,1355.6 0.00024 23,333 28,4596.7 0.00039 17,179 39,9307.1 0.00032 22,188 40,9558.4 0.00047 17,872 5,3008.2 0.00036 22,778 136,192 -

6.7 0.00039 17,179 19,485

40 106.2 0.00094 112,979 39175.5 0.00028 269,643 7,59754.0 0.00020 270,000 42,16666.7 0.00029 230,000 6,15050.0 0.00019 263,158 50,64788.9 0.00036 246,944 2,62546.7 0.00012 389,167 110,29695.5 0.00035 272,857 3,118102.2 0.00040 255,500 53646.7 0.00015 311,333 42,955

AA

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